On aspects of the physical realizability of perfectly matched absorbers for electromagnetic waves



[1] A discussion on the physical realizability of electromagnetic wave absorbers with perfectly matched impedances over arbitrary (i.e., doubly) curved, smooth surfaces is presented. The focus is on the analysis of the spectral characteristics (and their impact on the absorptive properties) of hypothetical material blueprints derived from the zero-reflection condition over such geometries.

1. Introduction

1.1. Background

[2] Continual advances on microfabrication technology in recent years have made possible, particularly in two dimensions, the controlled design of novel material composites on scales down to tens of angstroms. For many cases in the microwave/millimeter-wave range, the observed electromagnetic behavior at such length scales and above is still essentially dominated by properties at a classical level, i.e., those governed by Maxwell's equations not augmented by any inherently quantum effects (or amenable to be described, for all practical purposes, by equivalent classical models). Motivated by such advances, entirely new classes of novel materials with superior electromagnetic properties in the microwave/millimeter-wave range have been recently proposed and/or demonstrated, including materials (ferroelectrics) with voltage tunable dielectric permittivities and (comparatively) small loss tangents at room temperature [Sengupta and Klushens, 1998], ultrahigh dielectric materials (e.g., carbon nanotubes) [Garcia-Vidal et al., 1997; Slepyan et al., 1998], photonic bandgap materials (PBGs, also-called electromagnetic bandgap materials, or EBGs) [Yablonovitch, 1993; Sievenpiper et al., 1996; Pendry et al., 1998; Centeno and Felbacq, 2000; Kyriazidou et al., 2001], antiferroelectrics, (low-loss) magneto-dielectrics [Walser et al., 1998; Hansen and Burke, 2000], left-handed materials (also-called double negative media or backward-wave media) [Veselago, 1967; Smith et al., 2000; Pendry, 2000; Smith et al., 2001; Ziolkowski, 2001; Lindell et al., 2001; Engheta, 2001; Tretyakov, 2001], and low-k dielectrics [Shieh et al., 1998; Venugopal et al., 2000].

[3] One particular area in which microfabrication technology can have a definitive impact is on the design of wideband electromagnetic wave absorbers. A number of applications such as stealth technologies for radar cross-section (RCS) reduction and control, critically depend on the performance of antireflection composites [Rozanov, 2000]. Traditional designs of electromagnetic absorbers described in the open literature include Salisbury/Dallenbach screens and Jaumann absorbers [Chambers and Ford, 2000; Fante and McCormack, 1988], as well as ferrite-based absorbers [Amin and James, 1981]. The main limitations of classical designs are narrowband performance and their dependence on geometric properties of the coated surface. For instance, designs for planar surfaces show a degradation of the antireflection properties when placed over realistic curved surfaces, requiring extensive empirical corrections [Hashimoto and Mizokami, 1991]. In the last decade, there has been much interest in the possible use of chiral materials [Jaggard and Engheta, 1989; Jaggard et al., 1990; Bohren et al., 1992], and synthetic (bi)anisotropic materials with elementary (small) embedded scatterers of complex shape (helixes, omega shaped particles, etc.) [Brewitt-Taylor, 1994; Norgren, 1998; Tretyakov, 1998; Simovski et al., 2000]. This allows, in principle, greater flexibility in the design of electromagnetic wave absorbers through the addition of extra degrees of freedom on the constitutive parameters, e.g., magnetoelectric effects.

1.2. PML Absorbers: Planar Case

[4] Around 1994, the concept of a perfect matched layer (PML) absorber was introduced in computational electromagnetics [Berenger, 1994]. The PML corresponds to a reflectionless absorption layer (for all frequencies and incidence angles) developed as an absorbing boundary condition (ABC) for computational purposes [Berenger, 1994; Katz et al., 1994; Chew and Weedon, 1994; Liu and He, 1998]. Initially, such concept relied upon the introduction of matched artificial electric and magnetic conductivities and a splitting of the electromagnetic fields into subcomponents. Because of this, the resulting fields inside the PML layer were rendered nonphysical (non-Maxwellian). This was not of major relevance for the intended computational purposes per se, but it ruled out any physical interpretation for the reflectionless mechanism. Shortly thereafter, however, the PML concept has found an interesting dual formulation (Maxwellian PML) with a clear physical interpretation whereby the PML corresponds to particular frequency dependent material tensors (blueprints) equation image and equation image [Sacks et al., 1995; Gedney, 1996; Ziolkowski, 1997a, 1997b]. These tensors are material blueprints which exhibit large and matched imaginary parts for the responsible absorption mechanism.

[5] Because of the complex characteristics of the PML tensors, their actual physical realizability is (as expected) not a trivial matter. Nevertheless, physically realizable absorber concepts have been proposed to mimic the perfectly matched absorber behavior over a broad range of frequencies. For instance, a broad bandwidth absorbing material, based on the planar Maxwellian PML and hence potentially realizable with a proper engineering of materials, was introduced by Ziolkowski [1997a] and further studied by Ziolkowski [1997b]. This proposal is based upon a generalization of the Lorentz model for the polarization and magnetization fields that includes an extra time derivative of the driving fields. Moreover, it was then shown that this potentially realizable medium is both causal and passively absorbing. Another interesting proposal of a PML-like material is based on the use of uniaxial omega composites with higher order spatial dispersion effects [Tretyakov, 1998].

1.3. Curved PMLs

[6] The PML was originally derived in Cartesian coordinates (planar PML). However, the effectiveness of a planar PML is greatly reduced when applied to curved surfaces of electrically small radius of curvature. The PML concept was later extended to curvilinear coordinates [Chew et al., 1997; Maloney et al., 1997; Teixeira and Chew, 1997a; Collino and Monk, 1998; He and Liu, 1999; Hwang and Jin, 1999]. Although the first such extensions have dealt with non-Maxwellian formulations only, it was later shown that Maxwellian PMLs could also be obtained in curvilinear geometries [Teixeira and Chew, 1998]. Such Maxwellian PMLs correspond to anisotropic material tensors with inhomogeneous properties depending on the local geometry (principal curvatures) of the termination (coated) surface. From the existence of such extensions, the question naturally emerges whether or not they could provide blueprints for physically realizable absorbers over curved surfaces, similarly to the planar PML case.

[7] For classification purposes, we may divide the study of the physical realizability of these materials into three basic levels. The first level, denoted here blueprint level, amounts to investigating whether basic electromagnetic properties (i.e., causality, passivity) of the hypothetical material blueprints (frequency domain constitutive tensors) derived from the zero reflection conditions are satisfied [Ziolkowski, 1997b; Norgren and He, 1997]. The second level amounts to the search of time domain polarization models capable of approximating the desired response of the first level blueprints (ideal PML) at a given range of frequencies. At this second level, some compromise needs to be made on the frequency range under which such materials are expected to approximate the ideal PML behavior [Ziolkowski, 1997a; Ziolkowski and Auzanneau, 1997]. The third level consists on the investigation of specific material models (e.g., particulate composites with resonant, complex elementary scatterers) capable of furnishing the polarization responses derived at the second level [Auzanneau and Ziolkowski, 1998a, 1998b; Tretyakov, 1998, 2000]. Because of microfabrication constraints, additional compromises on the intended material performance are also present at this third level. (An additional level of analysis, apart from physical realizability studies but equally important for practical purposes, is related to the physical feasibility of such absorbers. This essentially amounts to the study of the structural (mechanical) and thermal conditions of any (physical level) proposed model and specific manufacturing constraints (e.g., thickness, density). A detailed numerical investigation at this level requires a multiphysics simulation approach.)

[8] As mentioned, for the planar PML, studies on the physical realizability of planar PML absorbers have been carried out at the different levels [Ziolkowski, 1997a; Tretyakov, 1998]. In this paper, we shall focus on the physical realizability of curved PML absorbers at the blueprint level.

2. Analysis

2.1. Curved PML Blueprints

[9] PML blueprints can be derived systematically for general geometries through a two step process. In the first step, a complexification of the metric of space is employed to map ordinary solutions of Maxwell's equations continuously into non-Maxwellian fields which exhibit a modified behavior (e.g., exponential decay) inside the PML [Teixeira and Chew, 1999a]. This complexification can be carried out by an analytic continuation of the spatial coordinates [Teixeira and Chew, 1998]. In the second step, another field mapping is employed. This second mapping transforms the non-Maxwellian (analytic continued) fields into a third set of fields, which are Maxwellian [Teixeira and Chew, 1999a]. This latter map explores the metric invariance of Maxwell's equations (in the sense of Deschamps [1981], Teixeira and Chew [1999b], and Bossavit [2001]), to recast the complexification of the metric as a change on the constitutive parameters. The end result is a set of Maxwell's equations in anisotropic media, where both the permeability and permittivity tensors are frequency dispersive and depend on the local metric coefficients of the PML surface (or, equivalently, the local radii of curvature of the PML).

[10] To describe these tensors, we attach a local orthogonal curvilinear coordinate system (ξ1, ξ2, ξ3) to a point P on the PML surface, where ξ1, ξ2 are tangent coordinates to the surface and ξ3 is the normal coordinate, such that ξ3 = 0 represents the PML interface itself and ξ3 > 0 represents points inside the PML. Using the convention eiωt, the PML blueprints for a doubly curved surface are written as

equation image
equation image

with the PML tensor [Teixeira and Chew, 1998]

equation image

In the above, s is the so-called complex stretching variable [Chew and Weedon, 1994], which has a frequency dependence on the form s(ω, ξ3) = a3) + i σ(ξ3)/ω, with a3) ≥ 1 and σ(ξ3) ≥ 0 in the PML (ξ3 > 0) (other functional dependencies of s in terms of ω are possible as long as they lead to an absorptive behavior). The factors equation imagei and hi, i = 1, 2 are the stretched and nonstretched local metric coefficients, respectively, given by hi = ri/r0i and equation image = equation imagei/r0i, i = 1, 2, and h3 = 1, where r0i1, ξ2), i = 1, 2, are the local principal radii of curvature at the point (ξ1, ξ2, 0). The radii of curvature are defined from the outside of the surface. Therefore, they are positive over concave surfaces and negative over convex surfaces. Moreover, ri = r0i + ξ3 and equation imagei = r0i + equation image3i = 1, 2, with

equation image

being the analytic continuation of the coordinate ξ3. The unit vectors equation imagei, i = 1, 2, are tangential to S at P along the principal lines of curvature, and equation image = equation image1 × equation image2 is the unit vector normal to S at this point. In terms of the local coordinates ξ1, ξ2, ξ3 of a local orthogonal system, we write equation imagei = (∂r/∂ξ)/∣∂r/∂ξi∣, i = 1, 2, where r is the position vector, and analogously for equation image. The imaginary part of the complex stretching variable s is responsible for the loss mechanism inside the PML region.

[11] The significance attached to equation (1) is that reflectionless absorption of incident waves on an curved surface can be obtained through by a hypothetical medium having properly chosen constitutive tensors. Strictly speaking, however, it is clear that the PML tensors given by equation (1) cannot represent the frequency behavior of real materials for all frequencies and, as a result, no physical significance can be attached to the absorber blueprint in equation (1) other than the (desired) constitutive behavior (objective function) to be approximated over some finite, preassigned bandwidth of interest. (For instance, in the limit ω → ∞ (in practice, this would correspond to the far ultraviolet for light element materials and X-ray frequencies for heavy element materials), a real material exhibits the following limiting approximate behavior for the permittivity (the fine structure is ignored for simplicity.) This has been the approach taken, for instance, in the work of Ziolkowski [1997a, 1997b] for physically realizable planar PMLs,

equation image

where ωp2 = Ne2/mϵ0, N is the number of atoms per unit volume, e and m are the electron charge and mass, respectively, and γ is a damping constant related to the collision rate. This approximate dependence is not exhibited by equation (1). In terms of the permeability, slowly decaying μij terms which decreases as 1/ω may be present up to optical frequencies for some anisotropic media such as ferromagnets. Also, some polarization modes related to the core electrons in the atoms may give rise to variations on the ϵ(ω) behavior other than above even at X-ray frequencies, but this is not germane to the main discussion here.)

2.2. Inhomogeneous Properties and Frequency Behavior

[12] In the case of planar PMLs, r0i → ∞, hi → 1, and equation imagei → 1, so that equation (1) reduces to the simpler form

equation image

[13] Meaningful microscopical time domain models can be obtained for equation (3) by using, for example, Lorentz models for the polarization equation image and magnetization equation image fields near the resonance (which implies, however, a narrowband performance if high absorption is desired), or, for more broadband behavior, through time-derivative Lorentz medium models for both equation image and equation image [Ziolkowski, 1997a].

[14] The more complicated nature of the curved PML blueprint equation image given by equation (1) when compared to the planar PML given by equation (3) is a consequence of the presence of the metric factors equation imagei/hi. The properties of the curved PML differ in two major ways from those of the planar PML:

  1. The curved PML tensor consist of rational functions of ω involving higher order polynomials than the planar case. As a result, higher order time-derivatives will need to be present in equivalent polarization and magnetization time domain models.
  2. The curved PML is, in general, an inhomogeneous medium in the transverse directions (ξ1, ξ2) because the metric factors in equation (1) depend on the local radii of curvature r0i1, ξ2), i = 1, 2.

[15] Both these facts lead to additional complexity for the necessary material engineering. A more fundamental aspect for the physical realizability, however, is the impact that the metric factors equation imagei/hi have on the spectral properties (when treating ω as a complex variable) of the PML blueprints. This is discussed next.

2.3. Spectral Properties; Limitations

[16] An important mathematical property to be observed by any constitutive parameter of a real passive material is the Kramers-Kronig (KK) relations [Landau et al., 1984]. The KK relations are often assumed to be a necessary condition for a material response to satisfy the primitive causality conditions (i.e., an effect cannot precede its cause). In terms of the constitutive relations, this is equivalent to the requirement equation image(t) = equation image(t) = 0 for t < 0. However, the KK relations are indeed only a sufficient condition for primitive causality, as will be discussed later.

[17] The KK relations applied to the PML tensor in equation (1) read as

equation image
equation image

for ω and ω′ real and where equation image(∞) ≡ limω→∞equation image(ω) is a real constant. In the above, PV denotes the Cauchy principal value, ℋ is a Hilbert transform, and equation image = −iRes[equation image(ω)]ω=0 = −i limω→0 ωequation image(ω). We call equation (4)generalized KK relations because they include an extra term in equation (4b) (residue contribution) due the pole at ω = 0, originating from (static) conductive losses in the PML [Teixeira and Chew, 1999c].

[18] In the case of the PML blueprints for curved surfaces, there is a major asymmetry between the spectral properties of equation image(ω) according to the local radii of curvature. This asymmetry directly impacts the validity of the generalized KK relations for equation image(ω). In order to study it more carefully, we first introduce some definitions.

  • Definition 1. A Concave or Planar (CoP) surface point is such that κ0i ≥ 0 on it, for i = 1, 2 where κ0i = 1/r0i are the local curvatures.
  • Definition 2. A Nonplanar, Nonconcave (NPNC) surface point is such that κ01 < 0 or κ02 < 0 on it.

[19] We have the following result for CoP surfaces.

  • Proposition 1.The PML tensorequation image(ω) at a CoP surface point satisfies the generalized KK relations.
    • Proof.Equation (4) are a consequence of the application of the Cauchy's theorem to the function (equation image(ω′) − equation image(∞))/(ω′ − ω) on the upper half of the complex ω plane (UHP), under the hypothesis that equation image(ω′) is analytic (holomorphic) there [Landau and Lifshitz, 1980]. Therefore, the proof just amounts to showing that equation image(ω) is analytic on UHP.

[20] In a concave or planar surface point P = (ξ1, ξ2, ξ3), we rewrite equation (1c) explicitly as

equation image


equation image
equation image
equation image

From the above, the singularities of equation image(ω) are found to be simple poles located at

equation image
equation image
equation image
equation image

For ξ3 ≥ 0, we have a3) ≥ 1 and σ(ξ3) ≥ 0 and therefore, from equation (2), b3) ≥ ξ3 ≥ 0, and Δ(ξ3) ≥ 0. For a concave surface r011, ξ2) > 0 and r021, ξ2) > 0 by definition and hence, from equations (7a) and (7b), both ω1, ω2, and ω3 are located on the lower half of the complex ω plane (LHP). For a planar surface, Λ11 = Λ22 = s and Λ33 = 1/s and hence the only poles are ω0 = 0 and ω4 = −iσ(ξ3)/a3). Therefore, for a planar or concave surface no singularities for equation image(ω) are present on the UHP, and equation image(ω) is analytic there. ♦

[21] A different conclusion is obtained for NPNC surfaces, as described by the next proposition.

  • Proposition 2.The PML tensorequation image(ω) at a NPNC surface point for which κ01 ≠ κ02violates the generalized KK relations.
    • Proof. In this case, the PML tensor again writes as equations (5)(6), and we still have, for ξ3 ≥ 0, that b3) ≥ ξ3 ≥ 0 and Δ(ξ3) ≥ 0. However, by definition r011, ξ2) < 0 or r021, ξ2) < 0 for a NPNC surface point and therefore at least one of the poles ω1, ω2 will be located on the UHP. As a result, equation image(ω) is not analytic on the UHP. If in the UHP, we denote these poles λ1 and λ2 (the latter may not exist). Given a equation image(ω′) tensor with simple poles on the UHP at ω′ = λk, k = equation image, the application of Cauchy's theorem to the associated tensor equation image(ω, ω′) = (equation image(ω′) − equation image(∞))/(ω′ − ω) on the UHP of ω′ gives
      equation image
      where ω is real, and the right hand side of equation (8) is a sum over all the residues of equation image(ω, ω′) on the UHP of ω′, N = 1 or N = 2 for equation image(ω) as in equations (5)(6). The last integral on the left hand side of equation (8) is carried out at a semicircle at infinity, C. Since limω′→∞equation image(ω, ω′) = 0, the integration over C vanishes. The two residue contributions on the left hand side of equation (8) are due to small indentations above the singularities of equation image(ω, ω′) on the real axis at ω′ = 0 and ω′ = ω.

[22] By writing

equation image

where equation image0(ω) is analytic at ω = 0, we have

equation image

and therefore

equation image

As a result, equation (8) becomes

equation image

Separating the real and imaginary part of the above we have

equation image
equation image

where have used, from equations (6)(7), that Res[equation image(ω′)]ω′=λk are constant and purely real tensors. Equation (10) are the counterpart to the generalized KK relations at a NPNC surface point. Both the residues of Λ11(ω′) and Λ22(ω′) at ω′ = ω1 and ω′ = ω2, respectively, may contribute to Res[equation image(ω′)]ω′=λk.

[23] Furthermore, from equations (5)(6) we have

equation image
equation image

where, from equation (2), a, b, σ and Δ are in general functions of position (ξ3 coordinate). Therefore, unless we have r01 = r02, the summation terms in the right-hand side of equation (10) are nonzero and Equation (10) do not reduce to equation (4). ♦

2.4. Time Domain Response

[24] As mentioned before, violation of the KK relations for a function equation image(ω) does not necessarily render a corresponding time domain transform equation image(t) noncausal. The actual behavior of equation image(t) depends on the particular choice for the Fourier inversion contour used to invert equation image(ω). A noncausal equation image(t) is obtained if an inverse Fourier contour along the real axis is used, as illustrated in Figure 1. On the other hand, a causal equation image(t) is obtained using a contour Cγ taken above all singularities, as illustrated in Figure 2. In the latter case, the inverse Fourier transform

equation image

can be closed, when t < 0, from the above by the semicircle at infinity CνC (Figure 2), which yields equation image(t) = 0 after applying Jordan's lemma. For t > 0, on the other hand, the contour Cγ can be closed from below, and application of Cauchy's theorem and Jordan's lemma yields

equation image

where we have assumed for simplicity [and according to equations (5)(6)], a equation image(ω) tensor in equation (12) having simple poles ω = λk, k = 1, …, N1 in the UHP (equation imagemk] > 0) and single poles ω = υk, k = 1, …, N2 elsewhere (equation imagemk] ≤ 0) as the only singularities. (The discussion remains essentially the same if multiple poles or branch point are present. In the first case, the summation should include terms of the form equation imagek(t)eiλkt and/or equation imagek(t)eiυkt, where equation imagek(t) is an (n − 1)-th order tensor polynomial and n is the multiplicity of the pole at ω = λk and/or ω = υk, respectively. In the second case, the summation should include terms of the form equation imagek(t)eiβkt, where βk is a branch point at ω = βk and equation imagek(t) depends on the difference of equation image(ω) along the two sides of the corresponding branch cut.) The summation at the left in equation (13) corresponds to the residue contributions from the UHP poles (UHP associated eigenmodes), while the summation at the right corresponds to the residue contributions from the poles elsewhere. Because equation imagemk] > 0, any term in the summation at the left is an unbounded function with exponential increase in time.

Figure 1.

Noncausal Fourier inversion contour Cη for Λ(ω) with singularities in the upper-half of the complex ω plane.

Figure 2.

Causal Fourier inversion contour Cγ for Λ(ω) with singularities in the upper-half of the complex ω plane. No singularity is enclosed by CγCν.

[25] In the previous section, we have determined that the PML blueprint equation image(ω) at a NPNC surface point with κ01 ≠ κ02 has at least one pole in the UHP. Because of this, the expression for susceptibility-like kernel equation image(t) reads as equation (13) with at least one term in the summation at the left. As a result, equation image(t) is unbounded function and such PML corresponds to an hypothetical medium which have internal sources of energy, i.e., artificially injects energy into the field (spurious active behavior), and not to an absorber medium. We also note here that for the degenerate case of NPNC surface points with κ01 = κ02, both residues in equation (11) are zero and hence the generalized KK relations are satisfied. Despite of that, a spurious active behavior is still present. In this case, this behavior is associated with the presence of zeros for equation image(ω) in the UHP [Aki and Richards, 1980; Landau and Lifshitz, 1980]. From equation (6c) with r01 = r02 = r0, we write

equation image

and we see that a double zero is present on the imaginary equation imagem[ω] axis at the point ω = −iσ/(r0 + b). In a NPNC surface r0 < 0 and this double zero is on the UHP. Hence, a similar analysis to the previous section can be made for the reciprocal kernel equation image−1(ω) with UHP singularities. (For simplicity, we have been focusing exclusively on the impact of the spectral properties of equation image on the constitutive equations. From the vector wave equation

equation image

we arrive, in the case of a PML medium, at

equation image

which depends explicitly both on equation image(ω) and its inverse.) On the contrary, for a CoP surface point, neither poles nor zeros are present on the UHP.

[26] The spurious behavior of PML blueprints on NCNP surfaces discussed above has been observed in time domain numerical simulations employing such hypothetical media in curvilinear grids [Teixeira and Chew, 1999c; Teixeira et al., 2001]. In these simulations, the behavior has manifested itself in terms of a strong dynamic instability established as soon as the incident wave impinges upon a NCNP PML region.

[27] It should be mentioned at this point that approximate PML blueprints can nevertheless still be obtained for NCNP geometries based on slight modifications from the exact PML blueprints. This could be done, for instance, by enforcing the poles and zeros in equation (6) to be on the LHP. Any such a posteriori modification would lead to blueprint absorbers suitable in principle for any geometries, but without reflectionless properties. In particular, enforcing Δ(ξ3) = 0 (planar PML) achieves this objective. Again, the performance of these blueprints depends on the local radii of curvature, with large radii ones exhibiting better performance (smaller reflections).

3. Conclusions

[28] We have discussed some theoretical aspects for the physical realizability of material blueprints for the reflectionless absorption of electromagnetic waves on general surface geometries.

[29] The results presented here have indicated that PML (i.e., reflectionless) absorber blueprints could not be established on all surface geometries. This is because the imposition of reflectionless conditions on some geometries (at NPNC surface points) is theoretically irreconcilable with absorptive effects. The resulting blueprints on such geometries exhibit a spurious active behavior (internal sources of energy) which manifest as unbounded time domain susceptibilities.

[30] This conclusion has been established against the backdrop of PML blueprint models that reduce to the usual planar PML in the limit of infinite radii of curvature [i.e., equation image = εequation image(ω) and equation image = μequation image(ω), with equation image given by equation (3)], which is an important condition to ensure compatibility in complex objects composed of both planar and curved surfaces. Consequently, this analysis does not rule out the existence of dissimilar reflectionless absorber blueprints for NCNP geometries (i.e., which do not reduce to the usual planar PML in the limit of infinite radii of curvature). Such possibility is currently under investigation.


[31] This work was presented in part at the 2001 URSI International Symposium on Electromagnetic Theory, Victoria, Canada, 13–17 May 2001.